Thursday, April 28, 2011

Are you smarter than a preservice middle school math teacher?

I gave my final for Teaching and Learning Middle Grades Mathematics on Wednesday; it had three parts. Learners completed the Teaching Math Workshop before class. The Learning Math Workshop was a collaborative effort completed during the first half of the time allotted for the final. During the last hour, learners worked individually on the Doing Math Workshop portion of the final.

The course is designed so the learners gradually take greater responsibility for their own learning. I find the workshop structure very supportive in attaining this goal. Consequently, the final Doing Math Workshop is little more than a basic set of instructions, followed by stripped down items from our text, and concluding with a self-evaluation. Check it out for yourself:

Doing Math Workshop (Final Exam)

TLW demonstrate his or her ability to model whit it means to do math.

Needs: 60 minutes, this workshop, and NCTM Process Standards (optional)

  • You can begin by reviewing NCTM Process Standards for Problem Solving, Representations, and Reasoning & Proof. [schema activation]
  • Next you need to read through the items provided and identify which will best support you in demonstrating what it means to do mathematics. [focus]
  • You can work on one, two, or three items to showcase your ability to problem solve, use representations, and reason mathematically. [activity]
  • Finally, you will need to go through your work and make explicit places where you are using the process standards. [reflection]
Problem Set

  1. 0.75 divided by 0.3
  2. x/9 = 7/12
  3. The Susa tablets from the vicinity of Babylon contain tables with relationships of the area of regular polygons to the square of the length of the side.
  4. The Type 9 pentagons discovered in February 1976 have four of the five sides congruent and angles measuring A, B, C, D, and E with the relationship 2E + B = 2D + C = 360. (The non-congruent side is between angles with measures D and E.)
  5. Compose two reflections over intersecting lines.
  6. In one form the Pythagorean Theorem states that if a right triangle has legs a and b and hypotenuse of length c, then squares may be constructed on the sides of the right triangle and a2 + b2 = c2. What happens if … ?

Self-evaluate your communication of your thinking using our 4 Cs + 1
Self-evaluation and evidence


What you did


How you did it


Why you did it


How it relates

Where it leads

Wednesday, April 27, 2011

Is direct instruction a better approach to teaching math?

I received the following in my email today:
An intriguing question that I often wonder about myself. Cambourne's research found that learning requires engagement, which means it depends on how engaging the lecture is to the learner. I clicked on the link hoping for some clarity. I was disappointed.

Here is a sampling from Paul E. Peterson’s article, Eighth-Grade Students Learn More Through Direct Instruction, reviewing the research:
As an instructor myself, I’ve had trouble making up my mind. I can cover a lot of ground in classes where lectures consume about two-thirds of the time. But those classes get less enthusiastic student evaluations than some smaller classes where students are encouraged to solve problems through discussion. I, too, like those problem-solving classes. They require less preparation and are easier to teach.
Before we more on, here are my reactions to this portion of the article. The first is nit-picky, but when Peterson says, “I can cover a lot of ground” it is a red flag for me. I, too, cover more material through lecture, but research shows that many students fail to cover the same ground or retain any memory of the landscape. Second, there is the statement, “I, too, like those problem-solving classes. They require less preparation and are easier to teach.” All I can say is that if planning a problem-solving lesson requires less preparation, then it is not really a problem-solving lesson.

Peterson continues:
So when Guido Schwerdt and Amelie Wuppermann of the University of Munich figured out a way to test empirically the relative value of the two teaching styles (see “Sage on the Stage,” research), it is worth trumpeting the findings. These analysts took advantage of the fact that the 2003 Trends in International Mathematics and Science Survey (TIMMS) not only tested a nationally representative sample of U.S. 8th graders in math and science, but also asked their teachers what percentage of class time was taken up by students “listening to lecture-style presentations” rather than either “working on problems with the teacher’s guidance” or “working on problems without guidance.” Teachers reported that they spent twice as much time on problem-solving activities as on direct instruction. In other words, U.S. middle-school teachers have drunk deep from the progressive pedagogical well.
It is important to note that the results are based on teachers' reporting instructional approaches and not direct observation. This is important because Stigler and Hiebert (1999) found that: “Although most U.S. teachers report trying to improve their teaching with current reform recommendations in mind, the videos show little evidence that change is occurring. Furthermore, when teachers do change their practice, it is often in only superficial ways.” (The Teaching Gap p. 12) This also comes from TIMSS data. However, there is no corroborating evidence in Schwerdt and Wuppermann’s study that supports teachers’ claims that they are using problem-solving approaches.

Furthermore, as the Learning Pyramid shows, not all direct-instruction methods are equal in their effectiveness. A demonstration (think aloud) would be a superior method to simply “covering content” through a traditional lecture. Again, Cambourne has shown that demonstrations are a necessary condition for learning but not sufficient. Learners must be given the opportunity to take responsibility for their learning and “give it a go.”

None of this seems to matter to Peterson who ends his review with, “Sadly, U.S. middle-school pedagogy is weighted heavily toward problem-solving.” In my opinion, what’s sad is that he would try to pass this research off as settling what is clearly a complex issue.

My reading of the Schwerdt and Wuppermann study suggests that it tries to answer the question, “Is traditional teaching really all that bad?” without considering how their methodology might misinterpret the data. I already discussed the problem with associating teacher reporting with using observable data. I am also concerned that they combined, “working on problems with the teacher’s guidance and working on problems without guidance” into the single problem-solving category.

Watch this lesson of a U.S. classroom where students are "working on problems with the teacher’s guidance" (from the original 1995 TIMSS research) and decide for yourself whether this teacher “drunk deep from the progressive pedagogical well.” (You will need to sign up for a password but it is free.) My answer is, “No,” but this would be categorized as problem-solving time in the Schwerdt and Wuppermann study.

Is traditional teaching really all that bad?” – based on this research, the jury is still out.

Friday, April 22, 2011

How will you organize your classroom of the future?

This past Wednesday was the last day for my Teaching and Learning Middle Grade Mathematics course. It was also the end of our unit on teaching mathematics, which meant developing an anchor chart to represent our current understanding (much like we did for doing math and learning math). Instead of having them summarize the content, I asked them to consider what their future classroom might look like given what we knew about doing, learning, and teaching mathematics.

The following video sums up the activity. This group of preservice teachers considered the role communication (learners placed in groups) and technology would play in their classroom but then they stopped and thought about their future learners. The result was surprising to me (the reason I like open-ended activities like this).

Cynically, I might think that they were trying to avoid the work, but they had already done the work. The final vision of their future classroom represented an authentic concern about meeting learners where they are at. The other groups agreed and I felt good about what they had learned this semester.

We spent the rest of the class watching a TED Talk and discussing the future of education. It wasn't the end-of-the-year party that a lot of classes were having, but it was a celebration. I love my job!

Thursday, April 21, 2011

Where do I turn?

Last week I attended the 2011 NCTM Conference in Indianapolis, Indiana. On Thursday, I learned how to use technology to make my thinking visible. And on Friday, I presented with some colleagues our efforts to offer middle school math and special education teachers job-embedded professional development. But it was the trip down and back that provided the teaching metaphor I wish to share in this post.

I picked up my colleague and co-presenter, Esther, at her house. She had one of those talking GPS gizmos and asked if I wanted to use it. I assured her that I had looked at the directions and it looked like a pretty straight shot. "Besides," I said, "those things annoy me. I don't want to be told where to turn all the time." That is how we began what was suppose to be a five hour trip.

We filled the first few hours discussing what was happening in our classes. I shared my struggles in getting my preservice teachers to make their thinking regarding the Process Standards explicit. In developing their think alouds, metacognitive memoirs, and reasoning recounts, they often miss opportunities to highlight whichever Standard they identify as the focus. Much of the time they express frustration with having to add this information because it interrupts the flow of their thinking. Esther reminded me that this was new to them and that they would need demonstrations and support. I agreed and shared my hope that the NCTM sessions I had selected would help with ideas that I could share with them.

Having addressed this issue, we checked to see how much longer it until Indianapolis. My Maps App said 3 hours, but that didn't make sense since we had been on the road for 3 hours already. It then became clear that my "pretty straight shot" had a major turn that I had missed near South Bend. Esther was gracious about my error and we made the necessary adjustment (with the help of the nice woman in the GPS gizmo with the Scottish accent).

We used Emma (the name Esther's girls gave the voice of the GPS guide) on the way home. Yes, she was annoying at times, but her purpose was clear: She interrupted our mindless progress with important information at critical points in the journey. This is where the teaching metaphor comes into play. Emma was what was missing from my preservice teachers' attempts to make their mathematical thinking visible. A mechanism to interrupt learners mindlessly following a teacher's thinking in such a way that they would take notice of how an expert made important decisions.

Please, do not take this metaphor too far. I am not suggesting that teachers take on the complete persona of a GPS guide - directing learners what to do at every turn. But when teachers are sharing their own thinking, describing their own learning path, it would not hurt to interrupt the flow of the lesson at critical points to ensure that learners do not miss an important point. Don't be afraid to be direct and annoying every now and then. It might help, though, to use a Scottish accent.

Tuesday, April 19, 2011

Why were they placed in pairs?

I wrote before how GVSU's Teacher Assisting semester is a fairly unique program. The Mathematics Department has gone one step further by placing pairs of Teacher Assistants [TAs] in classrooms (or as a cohort at a school). We have found that this approach has supported the TAs in numerous ways. The biggest benefit is that it helps them to break from the culture of isolation so prevalent in American teaching.

Today, the TAs did their final presentations. I wanted to try out a new app, MindNote, and created a concept map showing the connections made as they reflected on their experiences this semester. As you can see below, collaboration was important to many of the TAs.

What was interesting was the number of TAs who came into the semester skeptical of being placed with a peer. They thought that their experience would be diluted if they had to share a classroom. Instead, they found it a powerful and positive opportunity to improve their practice.

In a recent speech, Education Secretary Arne Duncan said we need to do more to attract great talent to teaching. Listening to the passion and commitment of these future teachers, I would respond that we do not need better teachers; we need to support better teaching. The best and the brightest are already here.

Sunday, April 17, 2011

How can I use technology to make my thinking visible?

Last week I attended the NCTM 2011 Annual Meeting and Exposition in Indianapolis. Whenever I attend conferences of this size I always like to choose some theme in order to select sessions that work together. This year, I decided to focus on ways to use technology to make thinking visible. The idea of using think alouds to model mathematical practices is a major goal of my Teaching and Learning Middle Grades Mathematics (MTH 329) course but it has been hard to put into practice because of our lack of experience. The NCTM Conference provided me with more experiences to share with my teachers in training.

The first session that I attended was Using Technology to Transform Students' Problem-Solving Experiences and Perspectives. The presentation was true to the title with many good examples used by the presenters in their classrooms. (I will try to post about these in more detail later.) What I found most interesting was the idea of using the green screen feature in Photo Booth to embed learners into their problem-solving efforts.

I decided to try this for myself, and the result is provided below. Unfortunately, for some reason, my work in the background kept fading in and out. This was especially bad whenever I moved my arms to gesture at particular points. That is why I used my eyes and head to focus the viewer's attention in the video:

In the second session, Using Screen-Capture Movies to Assess Quadrilateral Constructions in Sketchpad, presenters showed how they used Geometer's Sketchpad and Jing to share learners' efforts to explore quadrilateral properties. This session was especially appropriate since I assigned my MTH 329 class a Shape Maker Lab before I left for the conference. The final unit of the course is on teaching geometry and now I had another way to make my thinking visible in this content area.

The Jing software only allows for five minutes of video so it took several tries to get my timing down. I like that this feature forces the person doing the think aloud to consolidate his or her thinking to a reasonable length, but it does require some planning and practice. Using Sketchpad's redo command helped to make the initial construction go more quickly. I did the construction and then undid it without Jing running. And then I hit "redo" multiple times while Jing was recording until the initial construction was complete. Only the dynamic exploration at the end was completely "live" during the video:

Unable to display content. Adobe Flash is required.

Given that these think alouds using technology are my first attempts, I am happy with the results. However, I know that with more experience and support they can be improved. In fact, that is what we will be considering in MTH 329 tomorrow - glows and grows for these reasoning recounts. As always, your comments are welcome as well.

Saturday, April 16, 2011

What is job-embedded professional development?

The following is from the talk I gave at the NCTM 2011 Annual Meeting an Exposition. The title of the talk, Developing Teachers’ Algebraic Reasoning through Job-Embedded Professional Development, refers to our attempts to design a professional development experience based on teachers' immediate identified needs. This is the description we wrote almost a year ago:

Presenters will provide a model of job-embedded professional development used to support the teaching of algebraic reasoning in middle school.  We will explore how the use of context and representations enhanced content and pedagogical content knowledge to help support learners as they move from the concrete to the abstract. 
My co-presenters (Esther Huntzinger Billings and Mary Scott) and I struggled in developing the final presentation because things had not gone according to plan since we submitted the proposal last spring. Still, we were able to talk about what worked, what didn't, how we adjusted, and where we are headed. Under the circumstances, I think it went well. Here, see for yourself:

If you require more information about specific slides from the presentation, please use the comments to share your request.

Wednesday, April 13, 2011

How many now?

Hierarchical inclusion is the idea that whole numbers build by exactly one and that they nest within each other by this amount. (Fosnot & Dolk, pp. 36 - 37). Understanding hierarchical inclusion leads to the big ideas of decomposing & composing numbers and part-whole relationships. Consequently, hierarchical inclusion is an important concept learners need to experience and explore early on.

A few years ago, I wrote the book In Flew One More as a way to introduce hierarchical inclusion to early elementary learners. The book is a series of poems and illustrations that leverage learners desire to find patterns. Through reading and discussing this book, the idea of numbers building on by one is explicitly explored.

Given that April is national poetry month, I thought it was an appropriate time to share my book.

And here I am reading In Flew One More in my wife's first grade class this past Tuesday.

Finally, these are some cards related to the book that I made for a learning center on hierarchical inclusion.
I hope you enjoy National Poetry Month!

Saturday, April 9, 2011

Is there a Learning Museum?

Once again, Twitter has inspired me. This morning, I saw the following tweet from the National Math + Science Initiative:
Here's the money quote from the article
It also reinforces the emerging concept of “free choice” learning, which holds that people get most of their knowledge about science from someplace other than school or formal education.
What caught my attention was the phrase "free choice" learning. The idea that people take responsibility for their own learning is also found in Cambourne's work. This got me thinking about ways to change schools so that they more closely resemble museums where people learn, which led to this tweet:
The perspective that schools currently serve as "fact factories" comes from Sir Ken Robinson and Seth Godin.

I began to wonder what would be in a Learning Museum (not to be confused with Joe Bower's Museum of Education) and decided that it would have rooms dedicated to these topics:
  • What does learning look like?
  • How is a learner different than a student?
  • When and where has learning occurred?
  • Why is learning important?
  • What if I want to be a learner?
A google search found a Museum of Learning but no Learning Museum. Therefore, I decided to open a virtual Learning Museum. As curator, I will gather learning artifacts, but I cannot do it without your help. If you have any exhibits related to learning that you might like to share, please include links to them in the comments. I will try to gather them into the different rooms so people can stop by and improve their ability to learn. Again, from the article:
“The holy grail of science museums is not to provide someone all the knowledge they need, but to inspire them, to become a launching point,” said John Falk, an OSU professor of science education and national leader in the free-choice learning movement. 
The Learning Museum is now open. I hope it inspires you. Please visit often.

Thursday, April 7, 2011

What are the chances of a “Pig Out”? Part II

Last post, I shared with you the first half of an article I wrote with Dr. Mary Richardson (here is another article I wrote with Mary addressing fair games). Learners have just predicted the chances of rolling a "Pig Out" in the popular Pass the Pigs game. They are now ready to explore the experimental probability associated with this question.

The Activity (continued)
After learners share their subjective estimates, they are ready to test them by developing an experiment, as described on the Activity Worksheet.  In the activity, learners create possible plans for determining an estimate of the chances of a “Pig Out”.  Once plans have been suggested, the class discusses the plans, agrees on a plan to utilize, and carries out the plan.  One possibility would be to have each pair of learner toss the pig dice several times in order to see how often a “Pig Out” occurs. For example, within each pair, the following tasks are assigned:  one member tosses the pigs; and the other member records whether or not the pigs landed in the “Pig Out” position.  After one learner completes ten trials, the students switch tasks and ten more trials are conducted.  This provides partners with a total of twenty trials that can be used in determining an experimental probability.

Once the learners have conducted the experiment, we ask them to refine their original guesses.  When asked how they could be more certain of the probabilities, most respond that a longer experiment (more tosses) would result in more certainty.  However, they recognize that it is unrealistic for each of us to toss the pigs a very large number of times.  Instead, we decide to use the results from the entire class to get a better estimate for the probability.  We draw a number line on the whiteboard.  Each learner writes his/her own result from their tosses on a sticky note and posts the note appropriately on the line in order to form a dot plot.  After we have discussed the data for the whole class, everyone makes a new estimate for the probability.

Learners are asked to share their new probability estimates.  Most learners select the class median or class mode as their new estimate.  We estimate that a “Pig Out” is rolled one time in every five rolls.  Of course, differences in desktops, and the very nature of the experiment may produce varying results.

The important concept in this activity is that one cannot simply count outcomes for every object and assign a theoretical probability to any given outcome.  By exploring objects that cannot be assigned theoretical probabilities, learners seem to have a greater understanding of when a theoretical probability can be assigned.  Furthermore, learners recognize the need to generate data in order to obtain experimental probabilities
The exploration did not end there for us. We found several websites dedicated to Pass the Pigs. They contained online versions of the game and data collected in secondary classrooms. We used the data to update our experimental probabilities and look at new questions.

Sadly, the websites we found are no longer, but Wikipedia has data on rolling a single pig 11,954 times. Maybe you could use these to collect your own data to answer your own questions. For example, in order to win the game, you want to know when to pass the pigs. Therefore, the next question we asked was, "How long until we "Pig Out"? We came up with two ways to answer this question but I bet there are more.

Wednesday, April 6, 2011

What are the chances of a “Pig Out”? Part I

Several years ago, I wrote the following article with my colleague, Dr. Mary Richardson, for MCTM's Mathematics in Michigan (Volume 43, Number 1). Given that the issue is no longer available, I thought it would be appropriate to share it here. Because of the article's length it has been edited and will be split between two posts.

Middle grades learners should be provided ample opportunities to explore probability and statistics topics.  In particular, learners should be encouraged to collect, organize, and describe data, summarize the data, make predictions and conclusions based on the data, and test these predictions and conclusions (NCTM 2000).  [Update: this connects to CCSS 7.SP.C.6.] In the following activity, learners use the Pass the Pigs dice game to design a simulation experiment in order to estimate a probability. 

The Game
Pass the Pigs involves two tiny rubber pigs that players throw like dice (rules). Each pig can land in six different ways resulting in various point values.  A turn consists of a player taking as many rolls as he or she dares until: (1) deciding to stop and recording the total score for that turn, (2) a “Pig Out” is rolled and the player records a score of zero for that turn, or (3) an “Oinker” is thrown and all points accumulated in the game thus far by the player are lost.

The Activity
In this activity, learners explore the experimental probability of obtaining a “Pig Out” in the Pass the Pigs dice game.  The activity works best if learners have experience calculating measures of center, including mean, median, and mode.  Learners work in groups of two.  Each pair has one Pass the Pigs dice game and a flat table or desktop on which to work.  Each learner has a sticky note and a copy of the Activity Worksheet.

To begin the lesson, each pair of learners examines their pig dice.  We discuss the possible outcomes if the pig dice are simultaneously tossed.  Each learner is then asked to estimate the probability that a “Pig Out” will occur when playing Pass the Pigs.  Learners should write down and share their estimates and the reasoning behind them.  This leads into a discussion about different types of probability.  In addition to the two types of probability traditionally targeted in middle school (theoretical and experimental), this activity provides the opportunity to introduce learners to subjective probability.  Learners recognize that this is a situation for which there is no easy way to determine a theoretical probability and no data on hand for them to use to calculate an experimental probability.  Therefore, they must assign probabilities based upon what they think will happen (subjective probability).

To be continued... In the meantime, please estimate what you think is the probability that a "Pig Out" will occur and share it in the comments. Thank you.

Tuesday, April 5, 2011

"What if they come at you with a pointed stick?"

Brain Camborne has spent his career researching learning in natural settings in an effort to "find an educationally relevant theory of learning." This research led to Camborne identifying eight conditions present to support learning. The figure below comes from his book, The Whole Story, and shows how he envisions the conditions working together in a language arts classroom.

Engagement is central to these conditions; "the necessity for engagement underlies the whole plan, guiding and influencing all of the teacher's actions and interactions with the learners." Camborne found that three factors impacted learners' level of engagement (see figure to the right). 

In watching this Monty Python sketch, I was reminded of these factors and how they are too often ignored during instruction. [Sometimes it takes the extreme to become aware of the obvious.]

Why did only four cadets show up for the self-defense class? What might they expect from past lessons? Did they not show up because they felt they did not have the resources (guns, 16-ton weights, or tigers) needed to be successful? Were they uninterested in learning how to defend against assailants wielding fruit (instead of pointed sticks)? Or did they feel unsafe (and rightly so)?

More importantly, how do we unsure that our lessons address potential, purpose, and protection?

Monday, April 4, 2011

Where do you stand on teaching mathematics?

Today my Teaching and Learning Middle Grades Mathematics course begins their unit on teaching mathematics. This follows units on doing mathematics and learning mathematics. We begin this unit by taking a simile survey that asks respondents to complete the following:

Ideally, a mathematics teacher is like a(n):
a. Coach
b. Doctor
c. Entertainer
d. Gardener
e. News Broadcaster
f. Orchestra Conductor

  • Choose the simile that you believe best describes a mathematics teacher and explain your choice.
  • Choose the simile that does the worst job of describing a mathematics teacher and explain your choice.
  • Is there another simile that does a better job than these of describing a mathematics teacher? If there is, then what is it and what makes it better than these?

This survey follows similar simile surveys done for doing and learning mathematics. The doing mathematics unit had integers as one of the topics, so I asked my learners to rate the "doing" similes from -5 (the worst) to 5 (the best). We created a life-sized number line as a way to share and compare our choices. 

Rational number was the mathematical focus in the learning unit. For this survey, learners assigned their worst "learning" simile a 1 and then decided how many times better each other simile was than the worst one. The life-sized number line went from 0 to 1 this time. Learners computed the following ratio for each simile: simile rating to best simile rating. (i.e. The best simile was a one.) Again, we shared and compared our choices by standing on the number line.

The teaching mathematics unit concentrates on geometry. Today, I will ask them to stand in one of four corners representing strongly agree, agree, disagree, or strongly disagree for each simile. It is certainly a simpler process than the others but it often results in some interesting discussions.
Ideally, a mathematics teacher is like an entertainer. Where do you stand?

Sunday, April 3, 2011

Is mathematics about product or process?

Several years ago, a group of mathematics educators at GVSU decided to address an issue prevalent in our course, Mathematics for Elementary Teachers I. Many preservice elementary teachers in this course viewed mathematics from a product perspective. They considered only the answer as important. In response, we attempted to be more explicit regarding the mathematical processes by revising the course using the NCTM Process Standards as a framework. Below is one of the units that we developed around the theme of polygons.
In an effort to model promising teaching practices while maintaining the focus on the Process Standards, we introduced learning centers addressing Communication, Connections, Problem Solving, Representations, and Reasoning & Proof. The following centers came from a unit with Pentominoes as the theme. The preservice teachers rotated between these centers over the course of a two-hour class at the end of the unit.
Most of the ideas for the centers came from articles in Teaching Children Mathematics - NCTM's journal supporting improvement of pre-K-6 mathematics education. The content foci for the Pentominoes Unit were Measurement and Data Analysis and Probability. Some of the centers were closely related to the activities done during the unit, such as the Connection Center. Others, like the Reasoning & Proof Center, were new contexts that applied ideas found in previous lessons.

The preservice teachers looked forward to these centers at the end of each unit and were grateful for the instructional modeling of mathematics learning centers. Many had never experienced anything but traditional mathematics instruction in their K-12 experience. Still, I am not sure that one or two courses focusing on the Process Standards is enough to provide a more balanced perspective on what it means to do mathematics. After all, the process is as important as the product.

Saturday, April 2, 2011

Are you ready for "something completely different?"

My wife gave me The Complete Monty Python's Flying Circus for my 48th birthday. I just got around to watching the first episode and saw the following sketch. It demonstrates perfectly what happens when we fail to use formative assessment to inform planning and instruction.

It is important to understand that formative assessment does not need to be in the form of a test. We can gather data simply by observing. Imagine how this Italian class might have looked if the teacher had really seen the students during the lesson and adjusted his instruction accordingly. In the Design Thinking methodology, this is a part of the first phase - Empathy.

This reminds me of an article by Dr. Rheta Rubenstein titled, "The Learning of Mathematics: Perspectives of Different Researchers." (It comes from our text, Teaching and Learning Middle Grades Mathematics.) Under the section on Vygotsky, she includes the following diagram attributed to Dr. Lauren Resnick.

Vygotsky in Venn Diagram Form

As Terry Jones's character discovers, when teachers use scripted lessons that do not take learners' current level of understanding into account very little learning actually occurs.